The best time to go to Russia depends on what you want to do there...
Doing the Trans-Siberian...
Anytime is good really, but it's recommended to go either in Summer (June - September) or winter (December - March). Then you'll be able to participate fully. The best stops along the way are Novosibirsk, Krasnoyarsk, Irkutsk, Ulan-Ude. And you should definitely visit Baikal. If you've got time, take a bus to Olkhon island -- as wild and calm as it can get and (surprise!) very comfortable too (lots of accommodation options, good food, friendly locals, etc.)
Want to party...
You need to go either in May-June, September or December. This is the best time, lots of parties, exhibitions, events, and stuff like that. The best destinations are Moscow, St. Petersburg, Rostov-na-Donu, Volgograd, Samara.
Want to enjoy sightseeing...
Anytime except November is perfect. November is rainy and cold, ugly and disgusting. Don't go to Russia in November. The best place is St. Petersburg, Golden Ring (small towns above Moscow), Novgorod (beautiful ancient town between Moscow and St. Pete), and of course the whole of Trans-Siberian.